# Definition:Sigma-Algebra Generated by Collection of Subsets

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## Contents

## Definition

Let $X$ be a set.

Let $\mathcal G \subseteq \powerset X$ be a collection of subsets of $X$.

## Definition 1

The **$\sigma$-algebra generated by $\mathcal G$**, $\map \sigma {\mathcal G}$, is the smallest $\sigma$-algebra on $X$ that contains $\mathcal G$.

That is, $\map \sigma {\mathcal G}$ is subject to:

- $(1): \quad \mathcal G \subseteq \map \sigma {\mathcal G}$
- $(2): \quad \mathcal G \subseteq \Sigma \implies \map \sigma {\mathcal G} \subseteq \Sigma$ for any $\sigma$-algebra $\Sigma$ on $X$

## Definition 2

Then the **$\sigma$-algebra generated by $\mathcal G$**, $\map \sigma {\mathcal G}$, is the intersection of all $\sigma$-algebras on $X$ that contain $\mathcal G$.

### Generator

One says that $\mathcal G$ is a **generator** for $\map \sigma {\mathcal G}$.

Also, elements $G$ of $\mathcal G$ may be called **generators**.

## Also denoted as

Variations of the letter "$M$" can be seen for the $\sigma$-algebra generated by $\mathcal G$:

- $\map {\mathcal M} {\mathcal G}$
- $\map {\mathscr M} {\mathcal G}$

## Also see

- Existence and Uniqueness of Sigma-Algebra Generated by Collection of Subsets, where it is shown that $\map \sigma {\mathcal G}$ always exists, and is unique

## Sources

- 1984: Gerald B. Folland:
*Real Analysis: Modern Techniques and their Applications*: $\S 1.2$ - 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $3.2 \ \text{(ii)}$